A273719 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the peaks (n>=2, k>=1).

1, 1, 1, 3, 1, 1, 8, 3, 1, 1, 21, 9, 3, 1, 1, 55, 27, 10, 3, 1, 1, 144, 82, 33, 11, 3, 1, 1, 377, 251, 110, 39, 12, 3, 1, 1, 987, 770, 368, 139, 45, 13, 3, 1, 1, 2584, 2358, 1229, 495, 169, 51, 14, 3, 1, 1, 6765, 7191, 4085, 1755, 632, 200, 57, 15, 3, 1, 1

Offset: 2

Author

Emeric Deutsch, Jun 01 2016

Keywords

• nonn
• tabl

Comments

Number of entries in row n is n-1.

Sum of entries in row n = A082582(n).

T(n,1) = A088305(n-2) = F(2n-4) where F(n) are the Fibonacci numbers A000045.

Sum(k*T(n,k), k>=0) = A273720(n).

Examples

			Row 4 is 3,1,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding drawings show that they have 3,1,1,2,1 horizontal steps in their peaks.
Triangle starts
1;
1,1;
3,1,1;
8,3,1,1;
21,9,3,1,1
		

Links

• Alois P. Heinz, Rows n = 2..150, flattened
• A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A000045, A088305, A082582, A273720.

Programs

Maple

eq := G = z^2*s+z*(G-z^2*s/(1-z*s)+z^2*s^2/(1-z*s))+z*G+z^2*G+z*G*(G-z^2*s/(1-z*s)+z^2/(1-z)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 16 do seq(coeff(P[n], s, j), j = 1 .. n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t, h) option remember; expand(
      `if`(n=0, (1-t)*z^h, `if`(t<0, 0, b(n-1, y+1, 1, 0))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0)*z^h)+
      `if`(y<1, 0, b(n-1, y, 0, `if`(max(h, t)>0, h+1, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n-1))(b(n, 0$3)):
seq(T(n), n=2..16);  # Alois P. Heinz, Jun 06 2016

Mathematica

b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n == 0, (1 - t)*z^h, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]*z^h] + If[y < 1, 0, b[n - 1, y, 0, If[Max[h, t] > 0, h + 1, 0]]]]]; T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 1, n - 1}]][b[n, 0, 0, 0]];  Table[T[n], {n, 2, 16}] // Flatten (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz *)

Formula

G.f.: G(s,z), where s marks number of horizontal steps in the peaks and z marks semiperimeter, satisfies the equation given in the Maple program.

G.f.: G(w,z), where w marks number of horizontal steps in the peaks and z marks semiperimeter, satisfies eq. (7) of the Blecher et al. reference, where one has to set x = z and y = z.

The trivariate g.f. G = G(t,s,z), where t marks number of peaks, s marks number of horizontal steps in the peaks, and z marks semiperimeter, satisfies z*(1-z)*(1-s*z)*G^2-(1-3*z-s*z+z^2+3*s*z^2-s*z^3+t*s*z^3-t*s*z^4)*G + t*s*z^2*(1-z)^2 = 0.